[Wiener's Tauberian Theorem][1] says that linear combinations of translations of a function $f$ are dense in $L^1$ if and only if the zero set of the Fourier transform of $f$ is empty. This is an old theorem, from 1932(ish). I am interested in generalizations such as: When can a positive $L^1$ function approximated by _nonnegative_ linear combinations of translations of $f\ge 0$? What about convex combinations for approximating a density function?

I am looking for any modern reference covering these types of extensions, or other interesting extensions of this theorem. 


  [1]: https://en.wikipedia.org/wiki/Wiener%27s_tauberian_theorem